# A singular value decomposition updating algorithm for subspace tracking

A summary of the subspace tracking techniques is given in H. Viberg, in an article entitled “Two decades of array signal processing research,” IEEE Signal Processing Mag., vol. An alternative MIMO transmission techniques commonly referred to as V-BLAST uses a cancellation and nulling scheme requiring multiple instances of Moore-Penrose pseudo-inverses of a modified channel matrix H This approach is described e.g. Instead, the method uses the received vector r and the signal vector s or its estimate to update initial estimates of a) the left and right singular vectors, and b) the corresponding singular values, and to output the updated estimates thereof; the initial estimates are provided to the method as an input or are otherwise known.The method can be applied to any system that can be described by the linear matrix equation (1) with a transfer matrix H that is either constant or varies in dependence on time or some other parameter, e.g.

wherein the step of determining updated estimates of the singular values and singular vectors includes weighting of a contribution of the orthogonal projections of the input and output vectors into the updated estimates of the singular values and singular vectors with a weighting factor of less than 1.The left and right singular vectors form two orthogonal sets of vectors, which define M-dimensional receiver and K-dimensional transmitter spaces, respectively, with M typically equal to or exceeding K. (3) Therefore, knowledge of the SVD of the transfer matrix enables the establishment of up to K orthogonal independent communication channels between the receiver and the transmitter, and the deciphering of the transmitted signal at the receiver site. with frequency, the SVD needs to be periodically updated to maintain channel orthogonality and avoid inter-channel interference. In many applications, the MIMO channel and the corresponding transfer matrix changes slowly, so that the sets of the singular vectors do not change much between consecutive updates. 452-463, May 2003, disclosed an adaptive algorithm for use in MIMO wireless communication systems, wherein weight arrays are used at the transmitter and the receiver for establishing the orthogonal communications channels. In accordance with another aspect of this invention, the step of computing an orthogonal correction vector for each singular vector comprises the steps of recursively computing an orthogonal correction vector for each of the p left singular vectors and recursively computing an orthogonal correction vector for each of the p right singular vectors using the orthogonal projections of the output and input vectors onto initial estimates of the right and left singular vectors and the initial estimates of the singular values.Therefore, instead of performing a full SVD computation each time an SVD update is required, tracking techniques may be employed when the singular vectors and singular values at an nstep. Zeidler, in an article entitled “Feedback assisted transmission subspace tracking for MIMO systems,” IEEE Sel. The weight arrays are periodically jointly updated to adapt to changing transmission conditions. Wolniansky, entitled “Detection algorithm and initial laboratory results using VBLAST space-time communication architecture,” Elect. In accordance with another aspect of the invention, a method is provided for adaptive signal processing in multiple-input multiple-output communication systems comprising the steps of: a) transmitting information signal in multiple signal streams with a multiple-output transmitter; b) sampling multiple data streams received by a multiple-input receiver to form an output vector; c) obtaining an input vector related to the output vector and to a transfer matrix; d) obtaining initial estimates of left singular vectors, right singular vectors and singular values of a singular-value decomposition of the transfer matrix, e) determining updated estimates of the left singular vectors, right singular vectors and the singular values using the initial estimates thereof and projections of the input and output vectors on the initial estimates of the right and left singular vectors; f) using the updated estimates of the left and right singular vectors and the singular values for adaptively extracting the information signal from the multiple signal streams received by the receiver and/or for adaptively emitting the information signal in multiple signal streams by the transmitter.Singular value decomposition (SVD) of a matrix is a powerful mathematical tool used in many applications involving data processing, including data mining, dynamic system control, and communication systems.In particular, SVD is advantageously used in MIMO systems to determine orthogonal input-output channels over which information can be transmitted independently and without inter-channel interference, or to determine weight vectors and symbol detection order in systems employing successive interference cancellation.The method employs orthogonal projections of known output and input vectors onto initial estimates of left and right singular vectors to compute corrections to the initial estimates of the singular vectors and singular values and adaptively generate updated estimates thereof.

The method can be used for MIMO systems, including system of wireless MIMO communications employing adaptive transmitter and V-BLAST techniques.1.

In a typical MIMO communication system having K1 antennas at a receive site, each receive antenna can detect signal energy from each of the K transmit antennas, thereby providing K×M interfering signal paths between the receiver and the transmitter. discloses a wireless MIMO system employing the SVD at the transmitter to determine eigen-modes, i.e., spatial subchannels, of the MIMO channel and to derive a first set of steering vectors used to “precondition” modulation symbols. In accordance with the invention, a method is provided for updating singular values and singular vectors in a singular value decomposition (SVD) of a matrix transfer function relating an output vector to an input vector, the method comprising the steps of: a) obtaining initial estimates of the singular values and singular vectors; b) providing the output vector and the input vector; c) projecting the output vector and the input vector onto the initial estimates of the singular vectors to obtain orthogonal projections of the output and input vectors; and, d) determining updated estimates of the singular values and singular vectors from the orthogonal projections of the input and output vectors and the initial estimates of the singular values and the singular vectors.

Signal transmission in this system is commonly described by a matrix equation r=Hs ν (1) where s is an input vector of length K with element sreceive antenna, ν is a length M noise vector accounting for transmission and receiver noise, and H is a transfer matrix of size M×K accounting for multi-path interference and other linear transmission effects affecting the signal. The SVD is also performed at the receiver to derive a second set of steering vectors used to precondition the received signals, such that orthogonal symbol streams are recovered at the receiver. In accordance with one aspect of this invention, the method is for updating a subset of a full set of singular vectors, the subset comprising p left singular vectors and p right singular vectors, wherein p is an integer that is equal or less than min(M, K), wherein both K and M are integers and wherein K is a dimension of the input vector, and M is a dimension of the output vector.

comprising a step of estimating a rank of the matrix transfer function from one of the orthogonal projections of the output vector onto the left singular vectors and the orthogonal projections of the input vector onto the right singular vectors for detecting a change of said rank.

wherein the step of computing an orthogonal correction vector for each singular vector further comprises the step of computing (p1) orthogonal expansion coefficients for each singular vector from the orthogonal projections of the output and input vectors onto initial estimates of the right and left singular vectors using the initial estimates of the singular values.using the normalized residual input and output vectors scaled by the corresponding residual orthogonal expansion coefficients to compute the orthogonal correction vectors for the p left and p right singular vectors. Provisional Patent Application No: 60/534,189 filed Jan.

The algorithm employs one-sided transformations, and therefore provides a cheap alternative to earlier-developed updating algorithms based on two-sided transformations.